Reference Math & integrals
Jim Guenther1
1
Affiliation not available
October 12, 2022
Table of Contents
A) Useful integrals
B) Useful derivatives
C) Vectors and Tensors
1. Product formulas
2. Coordinate system conversions
3. Commutator relationships
D) Useful functions
1. Special functions
2. Associated Legendre Functions
3. Gamma Function
4. Bessel Functions
5. Hankel Functions
6. Spherical Harmonics
7. Airy Functions
8. Greens Function
9. Raleigh Formula (expansion of plane waves in terms of spherical waves)
10. Helmholtz Equation
11. Fourier Transform
E) Trigonometric functions and identities
1
INTEGRALS: Combination of exponentials and powers of x
Z ∞
2
x2n e−bx dx =
0
Z ∞
(2n)!
n!22n+1
2
x e
−bx2
0
Z ∞
2
0
Z ∞
x2
e− a2 dx =
0
1
2
r
1
dx =
4b
x4 e−bx dx =
3
8b2
r
r
π
b2n+1
1
2b
2
x e−bx dx =
0
Z ∞
r
π
b
√
π
3 π
=
5
b
8b 2
π
where a > 0
a
Sin and Cos Functions
Z
sin2 (kx) =
Z ∞
0
Z
x cos (ax) =
x sin (2kx)
−
2
4k
π
sin (kx)
=
x
2
1
x
cos (ax) + sin (ax)
a2
a
Exponentials with imaginary exponents
Z ∞
2
e−bx e−ikx dx =
r
−∞
Z ∞
−∞
2
eikx dx =
Z ∞
eikx
∗
eikx dx =
−∞
π − k2
e 4b
b
Z ∞
−∞
2
e−ikx
eikx dx = 1
Useful Derivatives
Vectors and Tensors
Scalar product or Dot product or Inner product:
This is a projection of one vector onto another. The result is a scalar value. The basic definition is:
A ◦ B = ax bx + ay by + az bz
You can also calculate the dot product using the angle between the two vectors, θ.
A ◦ B = |A| |B| cos θ
Further, you can calculate this angle using the equation below:
a b +a b +az bz
y y
Since we know that - cos θ = x x |A||B|
a b +ay by +az bz
Then θ = arccos x x |A||B|
Also note that the dot product of a unit vector - î, ĵ, k̂ - results in the component of the vector in the direction
of the unit vector.
Other notes:
Dot product of a vector with itself is equal to 1
A◦A=1
Dot product of orthogonal vectors is equal to zero
î ◦ ĵ = 0; ĵ ◦ k̂ = 0 and so fourth.
Cross Product
The Cross Product results in a vector that is perpendicular to the two original vectors. You can use the
right-hand rule to determine the direction of this new vector.
This equation occurs in physics in two situations:
1. Torque - application of a force onto a lever arm ~τ = ~r × F~
~
2. Movement of a charged particle in a magnetic field F~B = q~ν × B
Definition of Cross product:
~×B
~ = (Ay Bz − Az By ) î + (Az Bx − Ax Bz ) ĵ + (Ax By − Ay Bx ) k̂
A
You can also write this as a determinant:
3
î
~×B
~ = Ax
A
Bx
amp; ĵ
amp; Ay
amp; By
amp; k̂
amp; Az
amp; Bz
Note that the inverse relation is not the same, but is the inverse.
~×B
~ = −B
~ ×A
~
A
There are also useful equations for the angle between the two vectors:
~ sin θ
~ B
~×B
~ = A
For the magnitude of the resulting vector: A
Finally, cross products of the same unit vector equal 0, while cross products of two orthogonal unit vectors
will equal the third unit vector, or its negative (for reverse order cross products)
For example:
î × î = 0, ĵ × ĵ = 0
î × ĵ = 1, ĵ × k̂ = 1
k̂ × ĵ = −1, ĵ × î = −1
Triple Scalar Product
Start with the definition of Dot Product and Cross Products to get the Triple Scalar Product:
~ ◦ (B
~ × C)
~ = Ax (By Cz − Bz Cy ) î + Ay (Bz Cx − Bx Cz ) ĵ + Az (Bx Cy − By Cx ) k̂
A
An alternate presentation in determinant form:
Ax
~ ◦ (B
~ × C)
~ = Bx
A
Cx
amp; Ay
amp; By
amp; Cy
amp; Az
amp; Bz
amp; Cz
Finally note that the Triple Scalar Product is zero, if all three vectors lie in the same plane. For a non-zero
result, vector A must lie in a different plane than Vectors B and C.
~ ◦ (B
~ × C)
~ =0
A
Triple Vector Product
The Triple Vector Product (TVP) produces another vector in the same plane as the two vectors in parentheses
- B and C. Note that this new vector is a linear combination of vectors B and C.
In physics, this is a useful formula for Angular Momentum and Centripetal Acceleration
Simplified form for calculation:
~ × (B
~ × C)
~ =B
~ ◦ (A
~ × C)
~ −C
~ ◦ (A
~ × B)
~
A
The formula can also be written in determinant form
4
~ × (B
~ × C)
~ =
A
î
Ax
~ × C)
~ x
(B
amp; ĵ
amp; Ay
~ × C)
~ y
amp; (B
amp; k̂
amp; Az
~ × C)
~ z
amp; (B
or
~ × (B
~ × C)
~ =
A
î
amp; ĵ
amp; k̂
Ax
amp; Ay
amp; Az
(By Cz − Bz Cy ) amp; (Bz Cx − Bx Cz ) amp; (Bx Cy − By Cx )
Vectors and Tensors Section 2 - Operators using partial derivatives
Gradient
The gradient calculates how quickly a scalar field is changing over space. The result of a gradient is
a vector, even though the field is a scalar field. Here are equations for the gradient in three different
coordinate systems:
Cartesian coordinates
grad (ψ) = ∇ψ = î
∂ψ
∂ψ
∂ψ
+ ĵ
+ k̂
∂x
∂y
∂z
Spherical coordinates
grad (ψ) = ∇ψ = r̂
∂ψ
1 ∂ψ
1 ∂ψ
+ θ̂
+ φ̂
∂r
r ∂θ
rsinθ ∂φ
Cylindrical coordinates
grad (ψ) = ∇ψ = r̂
∂ψ
1 ∂ψ
∂ψ
+ φ̂
+ ẑ
∂r
r ∂φ
∂z
Divergence
Cartesian coordinates
~ ◦A
~ = ∂Ax + ∂Ay + ∂Az
∇
∂x
∂y
∂z
Spherical coordinates
~ ◦A
~ = 1 ∂ (r2 Ar ) + 1 ∂ (Aθ sinθ) + 1 ∂Aφ
∇
r2 ∂r
rsinθ ∂θ
rsinθ ∂φ
5
Curl
Cartesian coordinates
~ ×A
~=
∇
∂Az
∂Ay
−
∂y
∂z
î +
∂Ax
∂Az
−
∂z
∂x
ĵ +
∂Ay
∂Ax
−
∂x
∂y
k̂
or in matrix form:
î
~ ×A
~=
∇
∂
∂x
Ax
amp; ĵ
∂
amp; ∂y
amp; Ay
amp; k̂
∂
amp; ∂z
amp; Az
Spherical coordinates
~ ×A
~=
∇
1
∂(Aφ sin θ) ∂Aθ
1 1 ∂Ar
∂(rAφ )
1 ∂(rAθ ) ∂Ar
(
−
)r̂ + (
−
)θ̂ + (
−
)φ̂
r sin θ
∂θ
∂φ
r sin θ ∂φ
∂r
r
∂r
∂θ
Laplacian
Cartesian coordinates
∇2 φ =
∂2φ ∂2φ ∂2φ
+ 2 + 2
∂x2
∂y
∂z
Spherical coordinates
1 ∂
∇ ψ= 2
r ∂r
2
r
2 ∂ψ
∂r
+
∂2ψ
1
∂
∂ψ
1
(sin
θ
)
+
2
r2 sin θ ∂θ
∂θ
r2 sin θ ∂φ2
Laplacian is used in the Schrodinger equations:
2
~
2
Time dependent (TDSE): i~ ∂Ψ
∂t = − 2m ∇ Ψ + V Ψ
2
~
Time independent (TISE): − 2m
∇2 Ψ + V Ψ = EΨ
Coordinate system conversions
1) Spherical coordinates:
Spherical coordinates in terms of cartesian coordinates:
p
r = x2 + y 2 + z 2
θ = arccos √ 2 z 2 2
x +y +z
φ = arctan
y
x
Cartesian coordinates in terms of spherical coordinates:
6
x = r sin (θ) cos (φ)
y = r sin (θ) sin (φ)
z = r cos (θ)
Unit vectors in spherical coordinates are related to Cartesian unit vectors by the following:
r̂ = sin (θ) cos (φ) î + sin (θ) sin (φ) ĵ + cos (θ) k̂
θ̂ = cos (θ) cos (φ) î + cos (θ) sin (φ) ĵ − sin (θ) k̂
φ̂ = − sin (φ) î + cos (φ) ĵ
Figure 1: Spherical Coordinates
2) Cylindrical Coordinates:
Cylindrical coordinates in terms of cartesian coordinates:
p
r = x2 + y 2
φ = arctan xy
z=z
Cartesian coordinates in terms of cylindrical coordinates:
x = r cos (φ)
y = r sin (φ)
z=z
Unit vectors in cylindrical coordinates are related to Cartesian unit vectors by the following:
r̂ = cos (φ) î + sin (φ) ĵ
φ̂ = − sin (φ) î + cos (φ) ĵ
ẑ = ẑ
7
Commutators
Commutators are operators and the entire form is acting on some arbitrary function. Capital letters (i.e.
A, B & C) represent vectors. These are marked by a hat in the definition, but these hats have been dropped
for all the other equations. Small letters (i.e. c) represent constants.
h
i
Definition:
Â, B̂ = ÂB̂ − B̂ Â
Formulas:
[A, B] = − [B, A]
and
[A, A] = 0
[AB, C] = [A, C] B + A [B, C]
and
[A, BC] = [A, B] C + B [A, C]
[A + B, C + D] = [A.C] + [A, D] + [B, C] + [B, D]
Formulas with constants:
[A, c] = 0
[cA, B] = [A, cB] = c [A, B]
[a + A, b + B] = [A, B]
Useful Functions
1) Special functions
Levi-Civita tensor
ijk = +1,
ijk = 123, 231, 312
ijk = −1,
ijk = 132, 213, 321
ijk = 0,
otherwise
Kronecker delta
δnm = 1 when n = m
δnm = 0 when n 6= m
2) Associated Legendre function
Definition
l
Plm (x) = (−1)
1−x
Rodrigues formula to define Pl (x)
8
2
m2
d
dx
m
Pl (x)
Pl (x) =
1
2l l!
d
dx
l
l
x2 − 1
Table of the first few associated Legendre functions Plm (cos θ)
P00 (cos θ) = 1
P10 (cos θ) = cos θ
P20 (cos θ) = 21
P11 (cos θ) = − sin θ
3 cos2 θ − 1
P21 (cos θ) = −3 sin θ cos θ
P22 (cos θ) = 3 sin2 θ
P30 (cos θ) = 21 5 cos3 θ − 3 cos θ
P31 (cos θ) = − 32 sin θ 5 cos2 θ − 1
P32 (cos θ) = 15 sin2 θ cos θ
P33 (cos θ) = −15 sin θ 1 − cos2 θ
2) Gamma Function
Define the Gamma function as:
Z ∞
Γ(x) =
e−t tx−1 dt
(x > 0)
0
The recurrence function is:
Γ(x + 1) = xΓ(x)
From this we can generate these factorial expansions:
Γ(x + k) = (x + k − 1)(x + k − 2)(x + k − 3) ... xΓ(x)
and, setting x = 1, we get:
Γ(k + 1) = k(k − 1)(k − 2) ... 1 ... Γ(1) = k!
3) Bessel Functions
Bessel functions come from solutions of the Bessel equation:
x2
dy
d2 y
+x
+ x2 − a2 y = 0
dx2
dx
When a is an integer, these equations are called cylindrical functions or cylindrical harmonics. These
are represented by capital letters. The first kind is called a Bessel function, while the second kind is called
a Neumann or Weber function
Jn = first kind and Nn or Yn = second kind
A second form of Bessel equations is called Spherical Bessel Equations. These occur when a is an
half-integer. These are the equations we use for quantum mechanics in spherically symmetrical systems e.g. Coulomb potential. jn = first kind and nn or yn = second kind
9
Definition and a few values of Spherical Bessel functions of the first kind:
l
jl (x) ≡ (−x)
1 d
x dx
l
sin x
x
j0 = sinx x
cos x
x
j1 = sin
x2 − x
j2 = x33 − x1 sin x − x32 cos x
Asymptotic form when x 1 is :
jl =
2l l!
xl
(2l + 1)!
Definition and a few values of Spherical Bessel functions of the second kind (i.e. Spherical Neumann functions):
l
1 d
cos x
l
nl (x) ≡ − (−x)
x dx
x
n0 = cosx x
x
sin x
n1 = − cos
x2 − x
n2 = − x33 − x1 cos x − x32 sin x
Asymptotic form when x 1 is :
(2l)! 1
nl (x) = − l
2 l! xl+1
4) Hankel Functions
Hankel functions are combinations of Bessel functions of the first and second kind, to change the solutions
to an exponential from sin-like and cos-like. There are two types of Hankel functions as defined by the two
equations below:
(1)
= jl (x) + inl (x)
(2)
= jl (x) − inl (x)
hl
hl
We will need asymptotic values, so the equations below only represent Hankel functions where r 1. Values
for Spherical Hankel functions - of the first kind:
(1)
hl
(1)
=
1
l+1
(−i) eix
x
ix
h0 = −i ex
(1)
h1 = − xi2 − x1
(1)
eix
h2 = − x3i3 − x32 − xi
eix
10
We will need asymptotic values, so the equations below only represent Hankel functions where r 1. Values
for Spherical Hankel functions - of the second kind:
(2)
hl
(2)
−ix
(2)
1
i
x2 − x
=
1 l+1 −ix
(i) e
x
h0 = i e x
e−ix
(2)
h2 = x3i3 − x32 − xi e−ix
h1 =
5) Spherical Harmonics
Solutions for the 3D Schrodinger equation can be represented as a product of the solutions for the radial
and angular parts, due to separation of variables - ψ (r, θ, φ) = R (r) Θ (θ) Φ (φ)
Spherical Harmonics represent the combination of angular solutions. First is Φm (φ)
Φm (φ) = √12π eimφ
Solutions for various values of m:
Φm (φ) = √12π
m=0
Φm (φ) = √1π cos mφ
Φm (φ) = √1π sin |m| φ
m>0
m<0
Next is the solution for Θ (θ)
q
(l−m)! m
Θlm (cos θ) = (2l+1)
2
(l+m)! Pl (cos θ)
Finally we have the combined solution as a spherical harmonic
s
Ylm (θ,
φ) =
(2l + 1) (l − m)! m
P (cos θ) eimφ
2
(l + m)! l
Next is a table of a few Spherical Harmonics
Y00 (θ, φ) = √14π
q
3
Y11 (θ, φ) = − 8π
(sin θ) eiφ
q
3
(cos θ)
Y10 (θ, φ) = − 4π
q
3
Y1−1 (θ, φ) = − 8π
(sin θ) e−iφ
q
3
1
5
2
Y20 (θ, φ) = − 4π
2 cos θ − 2
q
7
5
3
3
Y30 (θ, φ) = − 4π
2 cos θ − 2 cos θ
11
6) Airy Functions
TBD
7) Green’s Function
In general, the Green’s Function for a linear differential equation represents the “response” to a delta function
source.
More precisely, given a linear differential operator acting on the collection of distributions over a subset of
some Euclidean space, a Green’s function is any solution of
L G (x, s) = δ (x − s)
The motivation for defining such a function is widespread, but by multiplying the above identity by a
function and integrating with respect to s.
R
R
L G (x, s) f (s) ds = δ (x − s) f (s) ds = f (s)
This s due to the characteristic of the delta function.
This equation is especially useful when solving for L u (x) = f (x)
R
Where u (x) = G (x, s) f (s) ds
In Quantum Mechanics, this can be applied as a solution to the Time independent Schrodinger equation. First write the TISE in the basic form of the Helmholtz equation and then use the Green’s function
in the solution:
2
~
∇2 ψ + V ψ = Eψ
Basic TISE: − 2m
Rewritten form: (∇2 + k 2 )ψ = Q
√
With substitutions: k ≡
2mE
~2
and
Q = 2m
~2 V ψ
Now assume there is a function (Green’s function) that solves this equation for a δ function source
(∇2 + k 2 )G (r) = δ 3 (r)
Now express this in integral form:
R
ψ (r) = G (r − r0 ) Q (r0 ) d3 r0
Finding Green’s function involves complex math, but here is the resulting integral.
R∞
G (r) = 4π12 r −∞ s ksin(sr)
2 −s2 ds
Solving this requires contour integration, so the final value is as follows. Note that this can be different,
depending on how you choose to handle poles in the contour integration.
G (r) =
eikr
i ikr
ikr
iπe
−
−iπe
=
8π 2 r
4πr
12
8) Rayleigh’s formula
In scattering theory and partial wave analysis, we have an incident plane wave and a reflected spherical
wave. Rayleigh’s formula provides a way to relate these to each other and solve the wave functions in terms
of spherical waves only.
eikz =
∞
X
il (2l + 1) jl (kr) Pl (cos θ)
l=0
BACKGROUND: This comes from a general solution of the Schrodinger equation with V = 0.
written in the following form:
X
It is
[Al,m jl (kr) + Bl,m nl (kr)] Ylm (θ, φ)
l,m
The problem here is that eikz is finite at the origin, but the Neumann function, nl (kr), is not; sonl is
dropped. This equation is then expanded using the facts that z = r cos θ has no φ dependance, and
only m = 0 terms occur.
[REF. Griffiths, p383; and Arfken exercises 15.2.24 & 15.2.25]
When you introduce a phase shift, we can rewrite the scattering cross section in terms of δ as
P∞
2
σ = 4π
l=0 (2l + 1) sin (δl )
k2
9) Helmholtz Equation
TBD
10) Fourier Transform
TBD
E) Trigonometric functions and Identities
cos (A + B) = cos (A) cos (B) − sin (A) sin (B)
sin (A + B) = sin (A) cos (B) + cos (A) sin (B)
sin A sin B = 21 [cos (A − B) − cos (A + B)]
Euler identity relations:
eiπ + 1 = 0
eix = cos (x) + i sin (x)
cos (x) = e
ix
+e−ix
2
13
sin (x) = e
ix
−e−ix
2i
14